Theorem r0cld 22349

Description: The analogue of the T₁ axiom (singletons are closed) for an R₀ space. In an R₀ space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
r0cld	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 22336	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1129	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		fncnvima2 6834	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
5	3, 4	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
6		fvex 6686	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
7	6	elsn 4585	. . . . 5 ⊢ ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ (𝐹‘𝑧) = (𝐹‘𝐴))
8		simpl1 1187	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		simpr 487	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
10		simpl3 1189	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1	kqfeq 22335	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
12		eleq2w 2899	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑜))
13		eleq2w 2899	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑜))
14	12, 13	bibi12d 348	. . . . . . . 8 ⊢ (𝑦 = 𝑜 → ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
15	14	cbvralvw 3452	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
16	11, 15	syl6bb 289	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
17	8, 9, 10, 16	syl3anc 1367	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
18	7, 17	syl5bb 285	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
19	18	rabbidva 3481	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}} = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
20	5, 19	eqtrd 2859	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
21	1	kqid 22339	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22	21	3ad2ant1 1129	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23		simp2 1133	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ Fre)
24		simp3 1134	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
25		fnfvelrn 6851	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
26	3, 24, 25	syl2anc 586	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
27	1	kqtopon 22338	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28	27	3ad2ant1 1129	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29		toponuni 21525	. . . . . 6 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
30	28, 29	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∪ (KQ‘𝐽))
31	26, 30	eleqtrd 2918	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽))
32		eqid 2824	. . . . 5 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
33	32	t1sncld 21937	. . . 4 ⊢ (((KQ‘𝐽) ∈ Fre ∧ (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽)) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
34	23, 31, 33	syl2anc 586	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35		cnclima 21879	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
36	22, 34, 35	syl2anc 586	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
37	20, 36	eqeltrrd 2917	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113  ∀wral 3141  {crab 3145  {csn 4570  ∪ cuni 4841   ↦ cmpt 5149  ^◡ccnv 5557  ran crn 5559   “ cima 5561   Fn wfn 6353  ‘cfv 6358  (class class class)co 7159  TopOnctopon 21521  Clsdccld 21627   Cn ccn 21835  Frect1 21918  KQckq 22304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-map 8411  df-qtop 16783  df-top 21505  df-topon 21522  df-cld 21630  df-cn 21838  df-t1 21925  df-kq 22305

This theorem is referenced by:  nrmr0reg  22360